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Mirrors > Home > ILE Home > Th. List > brcogw | Unicode version |
Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
Ref | Expression |
---|---|
brcogw |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 989 | . 2 | |
2 | simpl2 990 | . 2 | |
3 | breq2 3981 | . . . . . 6 | |
4 | breq1 3980 | . . . . . 6 | |
5 | 3, 4 | anbi12d 465 | . . . . 5 |
6 | 5 | spcegv 2810 | . . . 4 |
7 | 6 | imp 123 | . . 3 |
8 | 7 | 3ad2antl3 1150 | . 2 |
9 | brcog 4766 | . . 3 | |
10 | 9 | biimpar 295 | . 2 |
11 | 1, 2, 8, 10 | syl21anc 1226 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wex 1479 wcel 2135 class class class wbr 3977 ccom 4603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-br 3978 df-opab 4039 df-co 4608 |
This theorem is referenced by: (None) |
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